IB DP Physics: HL复习笔记11.3.7 Discharge Calculations

Capacitor Discharge Equation

  • The time constant is used in the exponential decay equations for the current, charge or potential difference (p.d) for a capacitor discharging through a resistor
    • These can be used to determine the amount of current, charge or p.d left after a certain amount of time for a discharging capacitor
  • This exponential decay means that no matter how much charge is initially on the plates, the amount of time it takes for that charge to halve is the same
  • The exponential decay of current on a discharging capacitor is defined by the equation:

7.7.3-Current-Discharge-Equation_3

  • Where:
    • I = current (A)
    • I0 = initial current before discharge (A)
    • e = the exponential function
    • t = time (s)
    • RC = resistance (Ω) × capacitance (F) = the time constant τ (s)
  • This equation shows that the smaller the time constant τ, the quicker the exponential decay of the current when discharging
  • Also, how big the initial current is affects the rate of discharge
    • If I0 is large, the capacitor will take longer to discharge
  • Note: during capacitor discharge, I0 is always larger than I, as the current I will always be decreasing

7.7.3-Capacitor-Discharge-Graph-and-Equation

Values of the capacitor discharge equation on a graph and circuit

  • The current at any time is directly proportional to the p.d across the capacitor and the charge across the parallel plates
  • Therefore, this equation also describes the charge on the capacitor after a certain amount of time:

7.7.3-Charge-Discharge-Equation_2

  • Where:
    • Q = charge on the capacitor plates (C)
    • Q0 = initial charge on the capacitor plates (C)
  • As well as the p.d after a certain amount of time:

7.7.3-Voltage-Discharge-Equation_2

  • Where:
    • V = p.d across the capacitor (C)
    • V0 = initial p.d across the capacitor (C)

The Exponential Function e

  • The symbol e represents the exponential constant, a number which is approximately equal to e = 2.718...
  • On a calculator, it is shown by the button ex
  • The inverse function of ex is ln(y), known as the natural logarithmic function
    • This is because, if ex = y, then x = ln (y)
  • The 0.37 in the definition of the time constant arises as a result of the exponential constant, the true definition is:Worked Example

The initial current through a circuit with a capacitor of 620 µF is 0.6 A. The capacitor is connected across the terminals of a 450 Ω resistor.

Calculate the time taken for the current to fall to 0.4 A.

7.7.3-Current-Discharge-Equation-Worked-Example

 

 

 

 

 

 

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